Axisymmetric bending of semi-infinite thin cylindrical shell under radial edge shear or edge moment — exponential-trigonometric closed-form solution (beam-on-elastic-foundation analogy)
This calculator gives exact closed-form results for the axisymmetric bending response of a semi-infinite thin cylindrical shell subjected to an axisymmetric edge load at its free end. It covers two fundamental load types:
The governing equation reduces to the beam-on-elastic-foundation (Winkler foundation) differential equation, whose solution has the well-known exponential-trigonometric form e^{-\lambda x}(\cos\lambda x \pm \sin\lambda x), where \lambda = [3(1-\nu^2)/(R^2t^2)]^{1/4} is the decay factor and D = Et^3/[12(1-\nu^2)] is the shell bending stiffness.
For radial edge shear V_0 (semi-infinite shell, long-shell solution):
y_A = -V_0/(2D\lambda^3), \psi_A = V_0/(2D\lambda^2);
y(x) = y_A\,e^{-\lambda x}\cos\lambda x, M(x) = -(V_0/\lambda)\,e^{-\lambda x}\sin\lambda x
For edge moment M_0 (semi-infinite shell, long-shell solution):
y_A = M_0/(2D\lambda^2), \psi_A = -M_0/(D\lambda);
y(x) = y_A\,e^{-\lambda x}(\cos\lambda x - \sin\lambda x), M(x) = M_0\,e^{-\lambda x}(\cos\lambda x + \sin\lambda x)
Derived stresses at evaluation point x: \sigma_2 = yE/R (hoop membrane), \sigma'_1 = -6M/t^2 (meridional bending, outer surface), \sigma'_2 = \nu\sigma'_1, \tau = V/t (average shear).
Model limits (first-order axisymmetric bending theory for thin cylindrical shells): thin wall R/t > 10; semi-infinite ("long") shell \lambda l > 6; constant wall thickness; isotropic linear elastic; small deformations; axisymmetric loading (no \theta dependence). Results do not cover short-shell end coupling, transverse-shear deformation, or thick-wall effects. For R/t \le 10, \lambda l \le 6, or to add internal-pressure membrane action, use the short-shell solution or FE shell model.
Covered cases of this table:
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